Let's consider a straight beam of constant cross-section with a length embedded at one end and loaded at the other end with a tensile force P (Fig. 8.2, a). Under the influence of force P, the beam elongates by a certain amount, which is called complete, or absolute, elongation (absolute longitudinal deformation).
At any points of the beam under consideration there is an identical state of stress and, therefore, linear deformations (see § 5.1) for all its points are the same. Therefore, the value can be defined as the ratio of the absolute elongation to the initial length of the beam I, i.e. Linear deformation during tension or compression of beams is usually called relative elongation, or relative longitudinal deformation, and is designated.
Hence,
Relative longitudinal strain is measured in abstract units. Let us agree to consider the elongation strain to be positive (Fig. 8.2, a), and the compression strain to be negative (Fig. 8.2, b).
The greater the magnitude of the force stretching the beam, the greater, other things being equal, the elongation of the beam; how larger area cross section beam, the less elongation of the beam. Bars from various materials lengthen differently. For cases where the stresses in the beam do not exceed the proportionality limit (see § 6.1, paragraph 4), the following relationship has been established by experience:
Here N is the longitudinal force in the cross sections of the beam; - cross-sectional area of the beam; E - coefficient depending on physical properties material.
Considering that normal voltage in the cross section of the beam we get
The absolute elongation of a beam is expressed by the formula
that is, the absolute longitudinal deformation is directly proportional to the longitudinal force.
For the first time, the law of direct proportionality between forces and deformations was formulated (in 1660). Formulas (10.2)-(13.2) are mathematical expressions of Hooke’s law for tension and compression of a beam.
The following formulation of Hooke's law is more general [see. formulas (11.2) and (12.2)]: the relative longitudinal strain is directly proportional to the normal stress. In this formulation, Hooke's law is used not only in the study of tension and compression of beams, but also in other sections of the course.
The quantity E included in formulas (10.2)-(13.2) is called the modulus of elasticity of the first kind (abbreviated as elastic modulus). This quantity is a physical quantity material constant, characterizing its rigidity. The greater the value of E, the less, other things being equal, the longitudinal deformation.
We will call the product the stiffness of the cross section of the beam under tension and compression.
Appendix I shows the values of elastic modulus E for various materials.
Formula (13.2) can be used to calculate the absolute longitudinal deformation of a section of a beam of length only under the condition that the section of the beam within this section is constant and the longitudinal force N is the same in all cross sections.
In addition to longitudinal deformation, when a compressive or tensile force is applied to the beam, transverse deformation is also observed. When a beam is compressed, its transverse dimensions increase, and when stretched, they decrease. If the transverse size of the beam before applying compressive forces P to it is designated b, and after the application of these forces (Fig. 9.2), then the value will indicate the absolute transverse deformation of the beam.
The ratio is the relative transverse strain.
Experience shows that at stresses not exceeding the elastic limit (see § 6.1, paragraph 3), the relative transverse deformation is directly proportional to the relative longitudinal deformation, but has the opposite sign:
The proportionality coefficient in formula (14.2) depends on the material of the beam. It is called the transverse deformation ratio, or Poisson's ratio, and is the ratio of the relative transverse deformation to the longitudinal deformation, taken in absolute value, i.e.
Poisson's ratio, along with the elastic modulus E, characterizes the elastic properties of the material.
The value of Poisson's ratio is determined experimentally. For various materials it has values from zero (for cork) to a value close to 0.50 (for rubber and paraffin). For steel, Poisson's ratio is 0.25-0.30; for a number of other metals (cast iron, zinc, bronze, copper) it has values from 0.23 to 0.36. Approximate values of Poisson's ratio for various materials are given in Appendix I.
When tensile forces act along the axis of the beam, its length increases and its transverse dimensions decrease. When compressive forces act, the opposite phenomenon occurs. In Fig. Figure 6 shows a beam stretched by two forces P. As a result of tension, the beam lengthened by an amount Δ l, which is called absolute elongation, and we get absolute transverse contraction Δа .
The ratio of the absolute elongation and shortening to the original length or width of the beam is called relative deformation. In this case, the relative deformation is called longitudinal deformation, A - relative transverse deformation. The ratio of relative transverse strain to relative longitudinal strain is called Poisson's ratio: (3.1)
Poisson's ratio for each material as an elastic constant is determined experimentally and is within the limits: 
; for steel.
Within the limits of elastic deformations, it has been established that the normal stress is directly proportional to the relative longitudinal deformation. This dependency is called Hooke's law:
, (3.2)
Where E- proportionality coefficient, called modulus of normal elasticity.
Have an idea of longitudinal and transverse deformations and their relationship.
Know Hooke's law, dependencies and formulas for calculating stresses and displacements.
Be able to carry out calculations of the strength and stiffness of statically determined beams in tension and compression.
Tensile and compressive strains
Let us consider the deformation of a beam under the action of a longitudinal force F(Fig. 4.13).
Initial dimensions of the timber: - initial length, - initial width. The beam is lengthened by an amount Δl; Δ1- absolute elongation. When stretched, the transverse dimensions decrease, Δ A- absolute narrowing; Δ1 > 0; Δ A<0.
During compression, the following relation is fulfilled: Δl< 0; Δ a> 0.
In the strength of materials, it is customary to calculate deformations in relative units: Fig.4.13
Relative extension;
Relative narrowing.
There is a relationship between longitudinal and transverse deformations ε′=με, where μ is the transverse deformation coefficient, or Poisson’s ratio, a characteristic of the plasticity of the material.
End of work -
This topic belongs to the section:
Theoretical mechanics
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All topics in this section:
Axioms of statics
The conditions under which a body can be in equilibrium are derived from several basic provisions, applied without proof, but confirmed by experience and called axioms of statics.
Connections and reactions of connections
All laws and theorems of statics are valid for a free rigid body. All bodies are divided into free and bound. A body that is not tested is called free.
Determination of the resultant geometrically
Know the geometric method of determining the resultant system of forces, the equilibrium conditions of a plane system of converging forces.
Resultant of converging forces
The resultant of two intersecting forces can be determined using a parallelogram or triangle of forces (4th axiom) (Fig. 1.13).
Projection of force on the axis
The projection of force onto the axis is determined by the segment of the axis, cut off by perpendiculars lowered onto the axis from the beginning and end of the vector (Fig. 1.15).
Determination of the resultant system of forces by an analytical method
The magnitude of the resultant is equal to the vector (geometric) sum of the vectors of the system of forces. We determine the resultant geometrically. Let's choose a coordinate system, determine the projections of all tasks
Equilibrium conditions for a plane system of converging forces in analytical form
Based on the fact that the resultant is zero, we obtain: FΣ
Methodology for solving problems
The solution to each problem can be divided into three stages. First stage: We discard the external connections of the system of bodies whose equilibrium is being considered, and replace their actions with reactions. Necessary
Couple of forces and moment of force about a point
Know the designation, module and definition of the moments of a force pair and a force relative to a point, the equilibrium conditions of a system of force pairs. Be able to determine the moments of force pairs and the relative moment of force
Equivalence of pairs
Two pairs of forces are considered equivalent if, after replacing one pair with another pair, the mechanical state of the body does not change, that is, the movement of the body does not change or is not disrupted
Supports and support reactions of beams
Rule for determining the direction of bond reactions (Fig. 1.22). The articulated movable support allows rotation around the hinge axis and linear movement parallel to the supporting plane.
Bringing force to a point
An arbitrary plane system of forces is a system of forces whose lines of action are located in the plane in any way (Fig. 1.23). Let's take the strength
Bringing a plane system of forces to a given point
The method of bringing one force to a given point can be applied to any number of forces. Let's say h
Influence of reference point
The reference point is chosen arbitrarily. An arbitrary plane system of forces is a system of forces whose line of action is located in the plane in any way. When changing by
Theorem on the moment of the resultant (Varignon’s theorem)
IN general case an arbitrary plane system of forces is reduced to the main vector F"gl and to the main moment Mgl relative to the selected center of reduction, and gl
Equilibrium condition for an arbitrarily flat system of forces
1) At equilibrium, the main vector of the system is zero (=0).
Beam systems. Determination of support reactions and pinching moments
Have an idea of the types of supports and the reactions that occur in the supports. Know the three forms of equilibrium equations and be able to use them to determine reactions in the supports of beam systems.
Types of loads
According to the method of application, loads are divided into concentrated and distributed. If the actual load transfer occurs on a negligibly small area (at a point), the load is called concentrated
Moment of force about a point
The moment of a force about an axis is characterized by the rotational effect created by a force tending to rotate a body around a given axis. Let a force be applied to a body at an arbitrary point K
Vector in space
In space, the force vector is projected onto three mutually perpendicular coordinate axes. Vector projections form edges rectangular parallelepiped, the force vector coincides with the diagonal (Fig. 1.3
Bringing an arbitrary spatial system of forces to the center O
A spatial system of forces is given (Fig. 7.5a). Let's bring it to the center O. The forces must be moved in parallel, and a system of pairs of forces is formed. The moment of each of these pairs is equal
Some definitions of the theory of mechanisms and machines
With further study of the subject of theoretical mechanics, especially when solving problems, we will encounter new concepts related to the science called the theory of mechanisms and machines.
Point acceleration
Vector quantity characterizing the rate of change in speed in magnitude and direction
Acceleration of a point during curvilinear motion
When a point moves along a curved path, the speed changes its direction. Let us imagine a point M, which, during a time Δt, moving along a curvilinear trajectory, has moved
Uniform movement
Uniform motion is motion at a constant speed: v = const. For rectilinear uniform motion (Fig. 2.9, a)
Uneven movement
With uneven movement, the numerical values of speed and acceleration change. Equation of uneven motion in general view is the equation of the third S = f
The simplest motions of a rigid body
Have an idea of translational motion, its features and parameters, and the rotational motion of a body and its parameters. Know the formulas for determining parameters progressively
Rotational movement
Movement in which at least the points of a rigid body or an unchanging system remain motionless, called rotational; a straight line connecting these two points,
Special cases of rotational motion
Uniform rotation ( angular velocity constant): ω = const. The equation (law) of uniform rotation in this case has the form: `
Velocities and accelerations of points of a rotating body
The body rotates around point O. Let us determine the parameters of motion of point A, located at a distance r a from the axis of rotation (Fig. 11.6, 11.7).
Rotational motion conversion
The transformation of rotational motion is carried out by various mechanisms called gears. The most common are gear and friction transmissions, as well as
Basic definitions
A complex movement is a movement that can be broken down into several simple ones. Simple movements consider translational and rotational. To consider the complex motion of points
Plane-parallel motion of a rigid body
Plane-parallel, or flat, motion of a rigid body is called such that all points of the body move parallel to some fixed one in the reference system under consideration.
Method for determining the instantaneous velocity center
The speed of any point on the body can be determined using the instantaneous center of velocities. In this case, complex movement is represented in the form of a chain of rotations around different centers. Task
Friction concept
Absolutely smooth and absolutely solid bodies do not exist in nature, and therefore, when one body moves over the surface of another, resistance arises, which is called friction.
Sliding friction
Sliding friction is the friction of motion in which the velocities of bodies at the point of contact are different in value and (or) direction. Sliding friction, like static friction, is determined by
Free and non-free points
A material point whose movement in space is not limited by any connections is called free. Problems are solved using the basic law of dynamics. Material then
The principle of kinetostatics (D'Alembert's principle)
The principle of kinetostatics is used to simplify the solution of a number of technical problems. In reality, inertial forces are applied to bodies connected to the accelerating body (to connections). d'Alembert proposal
Work done by a constant force on a straight path
The work of force in the general case is numerically equal to the product of the modulus of force by the length of the distance traveled mm and by the cosine of the angle between the direction of force and the direction of movement (Fig. 3.8): W
Work done by a constant force on a curved path
Let point M move along a circular arc and force F makes a certain angle a
Power
To characterize the performance and speed of work, the concept of power was introduced.
Efficiency
The ability of a body to do work when transitioning from one state to another is called energy. Energy is a general measure various forms mother's movements and interactions
Law of change of momentum
The quantity of motion of a material point is a vector quantity equal to the product of the mass of the point and its speed
Potential and kinetic energy
There are two main forms of mechanical energy: potential energy, or positional energy, and kinetic energy, or motional energy. Most often they have to
Law of change of kinetic energy
Let a constant force act on a material point of mass m. In this case, point
Fundamentals of the dynamics of a system of material points
A set of material points connected by interaction forces is called a mechanical system. Any material body in mechanics is considered as a mechanical
Basic equation for the dynamics of a rotating body
Let a rigid body, under the action of external forces, rotate around the Oz axis with angular velocity
Moments of inertia of some bodies
Moment of inertia of a solid cylinder (Fig. 3.19) Moment of inertia of a hollow thin-walled cylinder
Strength of materials
Have an idea of the types of calculations in the strength of materials, the classification of loads, internal force factors and resulting deformations, and mechanical stresses. Zn
Basic provisions. Hypotheses and assumptions
Practice shows that all parts of structures are deformed under the influence of loads, that is, they change their shape and size, and in some cases the structure is destroyed.
External forces
In the resistance of materials, external influences mean not only force interaction, but also thermal interaction, which arises due to uneven changes in temperature
Deformations are linear and angular. Elasticity of materials
Unlike theoretical mechanics, where the interaction of absolutely rigid (non-deformable) bodies was studied, in the strength of materials the behavior of structures whose material is capable of deformation is studied.
Assumptions and limitations accepted in strength of materials
Real Construction Materials, from which various buildings and structures are erected, are quite complex and heterogeneous solids with different properties. Take this into account
Types of loads and main deformations
During the operation of machines and structures, their components and parts perceive and transmit to each other various loads, i.e. force influences that cause changes in internal forces and
Shapes of structural elements
All the variety of forms is reduced to three types based on one characteristic. 1. Beam - any body whose length is significantly greater than other dimensions. Depending on the shape of the longitudinal
Section method. Voltage
Know the method of sections, internal force factors, stress components. Be able to determine types of loads and internal force factors in cross sections. For ra
Tension and compression
Tension or compression is a type of loading in which only one internal force factor appears in the cross section of the beam - longitudinal force. Longitudinal forces m
Central tension of a straight beam. Voltages
Central tension or compression is a type of deformation in which only longitudinal (normal) force N occurs in any cross section of the beam, and all other internal
Tensile and compressive stresses
During tension and compression, only normal stress acts in the section. Stresses in cross sections can be considered as forces per unit area. So
Hooke's law in tension and compression
Stresses and strains during tension and compression are interconnected by a relationship called Hooke's law, named after the English physicist Robert Hooke (1635 - 1703) who established this law.
Formulas for calculating the displacements of beam cross sections under tension and compression
We use well-known formulas. Hooke's law σ=Eε. Where.
Mechanical tests. Static tensile and compression tests
These are standard tests: equipment - a standard tensile testing machine, a standard sample (round or flat), a standard calculation method. In Fig. 4.15 shows the diagram
Mechanical characteristics
Mechanical characteristics of materials, i.e. quantities characterizing their strength, ductility, elasticity, hardness, as well as elastic constants E and υ, necessary for the designer to
Stresses and strains during tension and compression are related to each other by a linear relationship, which is called Hooke's law
, named after the English physicist R. Hooke (1653-1703), who established this law.
Hooke's law can be formulated as follows: normal stress is directly proportional to relative elongation or shortening
.
Mathematically, this dependence is written as follows:
σ = Eε.
Here E
– proportionality coefficient, which characterizes the rigidity of the timber material, i.e. its ability to resist deformation; he is called longitudinal modulus of elasticity
, or modulus of elasticity of the first kind
.
The elastic modulus, like the stress, is expressed in pascals (Pa)
.
Values E
for various materials are established experimentally, and their values can be found in the corresponding reference books.
So, for steel E = (1.96...2.16) x 105 MPa, for copper E = (1.00...1.30) x 105 MPa, etc.
It should be noted that Hooke's law is valid only within certain loading limits.
If we substitute the previously obtained values of relative elongation and stress into the formula of Hooke’s law: ε = Δl/l
,σ = N / A
, then you can get the following dependence:
Δl = N l / (E A).
Product of elastic modulus and cross-sectional area E × A , standing in the denominator, is called the section stiffness in tension and compression; it characterizes both the physical and mechanical properties of the timber material and geometric dimensions cross section of this beam.
The above formula can be read as follows: the absolute elongation or shortening of a beam is directly proportional to the longitudinal force and length of the beam, and inversely proportional to the stiffness of the beam's section.
Expression E A / l
called stiffness of the beam in tension and compression
.
The above formulas of Hooke's law are valid only for beams and their sections that have a constant cross-section, made of the same material and at a constant force. For a timber having several sections differing in material, section dimensions, longitudinal force, the change in the length of the entire beam is determined as the algebraic sum of lengthening or shortening of individual sections:
Δl = Σ (Δl i)
Deformation
Deformation(English) deformation) is a change in the shape and size of a body (or part of the body) under the influence of external forces, with changes in temperature, humidity, phase transformations and other influences, causing change positions of body particles. As the stress increases, the deformation may result in fracture. The ability of materials to resist deformation and destruction under the influence of various types loads are characterized mechanical properties these materials.
On the appearance of this or that type of deformation The nature of the stresses applied to the body has a great influence. Alone deformation processes are associated with the predominant action of the tangential component of stress, others - with the action of its normal component.
Types of deformation
According to the nature of the load applied to the body types of deformation divided as follows:
- Tensile strain;
- Compression strain;
- Shear (or shear) deformation;
- Torsional deformation;
- Bending deformation.
TO the simplest types of deformation include: tensile deformation, compression deformation, shear deformation. The following types of deformation are also distinguished: deformation of all-round compression, torsion, bending, which are various combinations of the simplest types of deformation (shear, compression, tension), since the force applied to a body subjected to deformation is usually not perpendicular to its surface, but directed at an angle , which causes both normal and shear stresses. Studying types of deformation Sciences such as solid state physics, materials science, and crystallography are involved.
IN solids, in particular metals, emit two main types of deformations- elastic and plastic deformation, the physical essence of which is different.
Shear is a type of deformation when only shear forces occur in cross sections.. Such a stressed state corresponds to the action on the rod of two equal, oppositely directed and infinitely close transverse forces (Fig. 2.13, a, b), causing a shear along a plane located between the forces.
Rice. 2.13. Strain and shear stress
Shearing is preceded by deformation - distortion right angle between two mutually perpendicular lines. At the same time, on the edges of the selected element (Fig. 2.13, V) tangential stresses arise. The amount of displacement of the faces is called absolute shift. The value of the absolute shift depends on the distance h between planes of action of forces F. The shear deformation is more fully characterized by the angle by which the right angles of the element change - relative shift:
. (2.27)
Using the previously discussed method of sections, it is easy to verify that only shear forces arise on the side faces of the selected element Q=F, which are the resultant tangential stresses:
Taking into account that shear stresses are distributed uniformly over the cross section A, their value is determined by the relation:
. (2.29)
It has been experimentally established that, within the limits of elastic deformations, the magnitude of tangential stresses is proportional to the relative shear (Hooke's law under shear):
Where G– modulus of elasticity under shear (modulus of elasticity of the second kind).
There is a relationship between the longitudinal elasticity and shear moduli
,
where is Poisson's ratio.
Approximate values of shear elasticity modulus, MPa: steel – 0.8·10 5 ; cast iron - 0.45 10 5; copper – 0.4·10 4; aluminum – 0.26·10 5; tires – 4.
2.4.1.1. Shear strength calculations
Pure shear in real structures is extremely difficult to implement, since due to the deformation of the connected elements, additional bending of the rod occurs, even with a relatively small distance between the planes of force action. However, in a number of structures, normal stresses in sections are small and can be neglected. In this case, the condition for the strength reliability of the part has the form:
, (2.31)
where are the permissible shear stresses, which are usually assigned depending on the value of the permissible tensile stress:
– for plastic materials under static load =(0.5...0.6) ;
– for fragile ones – =(0.7 ... 1.0) .
2.4.1.2. Shear stiffness calculations
They come down to limiting elastic deformations. By jointly solving expression (2.27)–(2.30), the magnitude of the absolute shift is determined:
, (2.32)
where is the shear stiffness.
Torsion
2.4.2.1. Constructing torque diagrams
2.4.2.2. Torsional Deformation
2.4.2.4. Geometric characteristics of sections
2.4.2.5. Strength and torsional rigidity calculations
Torsion is a type of deformation when a single force factor appears in cross sections - torque.
Torsional deformation occurs when a beam is loaded with pairs of forces, the planes of action of which are perpendicular to its longitudinal axis.
2.4.2.1. Constructing torque diagrams
To determine the stresses and deformations of the beam, a torque diagram is constructed showing the distribution of torques along the length of the beam. By applying the method of sections and considering any part in equilibrium, it will become obvious that the moment of internal elastic forces (torque) must balance the action of external (rotating) moments on the part of the beam under consideration. It is customary to consider a moment to be positive if the observer looks at the section under consideration from the side of the external normal and sees a torque T, directed counterclockwise. In the opposite direction, the moment is assigned a minus sign.
For example, the equilibrium condition for the left part of the beam has the form (Fig. 2.14):
– in cross section A-A:
– in cross section B-B:
.
The boundaries of the sections when constructing the diagram are the planes of action of torques.
Rice. 2.14. Design diagram of the beam (shaft) in torsion
2.4.2.2. Torsional Deformation
If a mesh is applied to the side surface of a rod with a round cross-section (Fig. 2.15, A) from equidistant circles and generators, and apply pairs of forces with moments to the free ends T in planes perpendicular to the axis of the rod, then with small deformation (Fig. 2.15, b) can be found:
Rice. 2.15. Torsional deformation pattern
· the generatrices of the cylinder turn into helical lines of large pitch;
· the squares formed by the grid turn into rhombuses, i.e. a shift of cross sections occurs;
· sections, round and flat before deformation, retain their shape after deformation;
· the distance between the cross sections practically does not change;
· one section rotates relative to another by a certain angle.
Based on these observations, the beam torsion theory is based on the following assumptions:
· cross sections of the beam, flat and normal to its axis before deformation, remain flat and normal to the axis after deformation;
Equally spaced cross sections rotate relative to each other by equal angles;
· the radii of cross sections do not bend during deformation;
· only shear stresses occur in cross sections. Normal stresses are small. The length of the beam can be considered unchanged;
· the material of the beam during deformation obeys Hooke's law in shear: .
In accordance with these hypotheses, the torsion of a rod with a circular cross-section is represented as the result of shears caused by the mutual rotation of the sections.
On a rod of circular cross-section with a radius r, sealed at one end and loaded with torque T at the other end (Fig. 2.16, A), let us denote the generatrix on the lateral surface AD, which under the influence of the moment will take the position AD 1. On distance Z from the embedding, select an element with length dZ. As a result of torsion, the left end of this element will rotate by angle , and the right end by angle (). Formative Sun element will take position B 1 C 1, deviating from the original position by an angle. Due to the smallness of this angle
The ratio represents the angle of twist per unit length of the rod and is called relative twist angle. Then
Rice. 2.16. Calculation scheme for determining stresses
when torsion of a rod of circular cross-section
Taking into account (2.33), Hooke's law under torsion can be described by the expression:
. (2.34)
Due to the hypothesis that the radii of circular cross sections do not bend, tangential shear stresses in the vicinity of any point of the body located at a distance from the center (Fig. 2.16, b), are equal to the product
those. proportional to its distance to the axis.
The value of the relative angle of twist according to formula (2.35) can be found from the condition that the elementary circumferential force () on an elementary area of size dA, located at a distance from the axis of the beam, creates an elementary moment relative to the axis (Fig. 2.16, b):
The sum of elementary moments acting over the entire cross section A, equal to torque M Z. Assuming that:
.
The integral represents purely geometric characteristic and is called polar moment of inertia of the section.
Lecture outline
1. Deformations, Hooke’s law during central tension-compression of rods.
2. Mechanical characteristics of materials under central tension and compression.
Let's consider a structural rod element in two states (see Figure 25):
External longitudinal force F absent, the initial length of the rod and its transverse size are equal, respectively l And b, cross-sectional area A the same along the entire length l(the outer contour of the rod is shown by solid lines);
The external longitudinal tensile force directed along the central axis is equal to F, the length of the rod received an increment Δ l, while its transverse size decreased by the amount Δ b(the outer contour of the rod in the deformed position is shown by dotted lines).
l Δ l
Figure 25. Longitudinal-transverse deformation of the rod during its central tension.
Incremental rod length Δ l is called its absolute longitudinal deformation, the value Δ b– absolute transverse deformation. Value Δ l can be interpreted as longitudinal movement (along the z axis) of the end cross section of the rod. Units of measurement Δ l and Δ b same as initial dimensions l And b(m, mm, cm). In engineering calculations, the following sign rule for Δ is used l: when a section of the rod is stretched, its length and value Δ increase l positive; if on a section of a rod with an initial length l internal compressive force occurs N, then the value Δ l negative, because there is a negative increment in the length of the section.
If absolute deformations Δ l and Δ b refer to initial sizes l And b, then we obtain relative deformations:
– relative longitudinal deformation;
– relative transverse deformation.
Relative deformations are dimensionless (as a rule,
very small) quantities, they are usually called e.o. d. – units relative deformations(For example, ε = 5.24·10 -5 e.o. d.).
The absolute value of the ratio of the relative longitudinal strain to the relative transverse strain is a very important material constant called the transverse strain ratio or Poisson's ratio(after the name of the French scientist)
As you can see, Poisson's ratio quantitatively characterizes the relationship between the values of relative transverse deformation and relative longitudinal deformation of the rod material when external forces are applied along one axis. The values of Poisson's ratio are determined experimentally and are given in reference books for various materials. For all isotropic materials, the values range from 0 to 0.5 (for cork close to 0, for rubber and rubber close to 0.5). In particular, for rolled steels and aluminum alloys in engineering calculations it is usually accepted, for concrete.
Knowing the value of longitudinal deformation ε (for example, as a result of measurements during experiments) and Poisson's ratio for a specific material (which can be taken from a reference book), you can calculate the value of the relative transverse strain
where the minus sign indicates that longitudinal and transverse deformations always have opposite algebraic signs (if the rod is extended by an amount Δ l tensile force, then the longitudinal deformation is positive, since the length of the rod receives a positive increment, but at the same time the transverse dimension b decreases, i.e. receives a negative increment Δ b and the transverse strain is negative; if the rod is compressed by force F, then, on the contrary, the longitudinal deformation will become negative, and the transverse deformation will become positive).
Internal forces and deformations arising in structural elements under the influence of external loads, represent a single process in which all factors are interconnected. First of all, we are interested in the relationship between internal forces and deformations, in particular, during central tension-compression of structural rod elements. In this case, as above, we will be guided Saint-Venant's principle: the distribution of internal forces significantly depends on the method of applying external forces to the rod only near the point of loading (in particular, when forces are applied to the rod through a small area), and in parts quite remote from the places
application of forces, the distribution of internal forces depends only on the static equivalent of these forces, i.e., under the action of tensile or compressive concentrated forces, we will assume that in most of the volume of the rod the distribution of internal forces will be uniform(this is confirmed by numerous experiments and experience in operating structures).
Back in the 17th century, the English scientist Robert Hooke established a direct proportional (linear) relationship (Hooke's law) of the absolute longitudinal deformation Δ l from tensile (or compressive) force F. In the 19th century, the English scientist Thomas Young formulated the idea that for each material there is a constant value (which he called the elastic modulus of the material), characterizing its ability to resist deformation under the action of external forces. At the same time, Jung was the first to point out that linear Hooke's law is true only in a certain region of material deformation, namely – during its elastic deformations.
In the modern concept, in relation to uniaxial central tension-compression of rods, Hooke’s law is used in two forms.
1) Normal stress in the cross section of a rod under central tension is directly proportional to its relative longitudinal deformation
, (1st type of Hooke's law),
Where E– the modulus of elasticity of the material under longitudinal deformations, the values of which for various materials are determined experimentally and are listed in reference books that technicians use when carrying out various engineering calculations; Thus, for rolled carbon steels, widely used in construction and mechanical engineering; for aluminum alloys; for copper; for other materials value E can always be found in reference books (see, for example, “Handbook on Strength of Materials” by G.S. Pisarenko et al.). Units of elastic modulus E the same as the units of measurement of normal stresses, i.e. Pa, MPa, N/mm 2 and etc.
2) If in the 1st form of Hooke’s law written above, the normal stress in the section σ express in terms of internal longitudinal force N and cross-sectional area of the rod A, i.e. , and the relative longitudinal deformation – through the initial length of the rod l and absolute longitudinal deformation Δ l, i.e., then after simple transformations we obtain a formula for practical calculations (longitudinal deformation is directly proportional to the internal longitudinal force)
(2nd type of Hooke's law). (18)
From this formula it follows that with increasing value of the elastic modulus of the material E absolute longitudinal deformation of the rod Δ l decreases. Thus, the resistance of structural elements to deformation (their rigidity) can be increased by using materials with higher elastic modulus values. E. Among the structural materials widely used in construction and mechanical engineering, they have a high elastic modulus E have steel. Value range E For different brands small steels: (1.92÷2.12) 10 5 MPa. For aluminum alloys, for example, the value E approximately three times less than that of steels. Therefore for
For structures with increased rigidity requirements, steel is the preferred material.
The product is called the rigidity parameter (or simply rigidity) of the cross section of the rod during its longitudinal deformations (the units of measurement of the longitudinal stiffness of the section are N, kN, MN). Magnitude c = E A/l is called the longitudinal stiffness of the rod length l(units of measurement of the longitudinal stiffness of the rod With – N/m, kN/m).
If the rod has several sections ( n) with variable longitudinal stiffness and complex longitudinal load (a function of the internal longitudinal force on the z coordinate of the cross section of the rod), then the total absolute longitudinal deformation of the rod will be determined by more general formula
where integration is carried out within each section of the rod of length , and discrete summation is carried out over all sections of the rod from i = 1 before i = n.
Hooke's law is widely used in engineering calculations of structures, since most structural materials during operation can withstand very significant stresses without collapsing within the limits of elastic deformations.
For inelastic (plastic or elastic-plastic) deformations of the rod material, the direct application of Hooke’s law is unlawful and, therefore, the above formulas cannot be used. In these cases, other calculated dependencies should be applied, which are discussed in special sections of the courses “Strength of Materials”, “Structural Mechanics”, “Mechanics of Solid Deformable Body”, as well as in the course “Theory of Plasticity”.